Review of the Canonical Equations

A block diagram for a generalized feedback system is repeated in Figure 6–1. This simple block diagram is sufficient to determine the stability of any system.

Figure 6–1. Feedback System Block Diagram

The output and error equation development is repeated below.

Combining Equations 6–1 and 6–2 yields Equation 6–3:

Collecting terms yields Equation 6–4:

Rearranging terms yields the classic form of the feedback equation.

Notice that Equation 6–5 reduces to Equation 6–6 when the quantity Aβ in Equation 6–5 becomes very large with respect to one. Equation 6–6 is called the ideal feedback equation because it depends on the assumption that Aβ >> 1, and it finds extensive use when amplifiers are assumed to have ideal qualities. Under the conditions that Aβ >>1, the system gain is determined by the feedback factor β. Stable passive circuit components are used to implement the feedback factor, thus the ideal closed loop gain is predictable and stable because β is predictable and stable.

The quantity Aβ is so important that it has been given a special name, loop gain. Consider Figure 6–2; when the voltage inputs are grounded (current inputs are opened) and the loop is broken, the calculated gain is the loop gain, Aβ. Now, keep in mind that this is a mathematics of complex numbers, which have magnitude and direction. When the loop gain approaches minus one, or to express it mathematically 1 ∠ –180°, Equation 6–5 approaches infinity because 1/0 ⇒ ∞. The circuit output heads for infinity as fast as it can using the equation of a straight line. If the output were not energy limited the circuit would explode the world, but it is energy limited by the power supplies so the world stays intact.

Figure 6–2. Feedback Loop Broken to Calculate Loop Gain

Active devices in electronic circuits exhibit nonlinear behavior when their output approaches a power supply rail, and the nonlinearity reduces the amplifier gain until the loop gain no longer equals 1∠ –180°. Now the circuit can do two things: first, it could become stable at the power supply limit, or second, it can reverse direction (because stored charge keeps the output voltage changing) and head for the negative power supply rail.

The first state where the circuit becomes stable at a power supply limit is named lockup; the circuit will remain in the locked up state until power is removed. The second state where the circuit bounces between power supply limits is named oscillatory. Remember, the loop gain, Aβ, is the sole factor that determines stability for a circuit or system. Inputs are grounded or disconnected when the loop gain is calculated, so they have no effect on stability. The loop gain criteria is analyzed in depth later.

Equations 6–1 and 6–2 are combined and rearranged to yield Equation 6–7, which gives an indication of system or circuit error.

First, notice that the error is proportional to the input signal. This is the expected result because a bigger input signal results in a bigger output signal, and bigger output signals require more drive voltage. Second, the loop gain is inversely proportional to the error.

As the loop gain increases the error decreases, thus large loop gains are attractive for minimizing errors. Large loop gains also decrease stability, thus there is always a tradeoff between error and stability.